A little extra snippet on just how much Dr Maynard loves prime numbers!!! kzfaq.info/get/bejne/o9uGlrOanNW7h40.html

"Terry Tao only beat me by one day." That's pretty badass, dude.

What sound does a drowning mathematician make? loglogloglogloglog

The US really needs 53 states, then we really could be "One Nation, Indivisible...."

All prime numbers except 2 are odd, this makes 2 the oddest prime.

The fact that maynard independently proved this conjecture within 1 day of Tao's collaborative effort is astounding. This kid is wicked smart.

"times a small constant c" *writes a tiny letter c*

Anyone else think James Maynard would be the best math teacher ever? He's so polite and enthusiastic

Dr Maynard talks about math with the kind of genuine excitement only a child would show, I loved every second of this video!

Dr Maynard has won the Fields Medal! Congratulations!

I just want to thank you guys for continuing to bring cutting edge maths into the public eye.

naughty brady using comic sans.

Near the start of this video I thought to myself, "I bet the answer involves a log function...". I had no idea how much I was going to end up laughing at the end.

I wonder if Matt gave this a go, and got it almost right...

"There are a prime gaps bigger than the number of atoms in the universe." Ultrafinitists:TRIGGERRED

Where's Ramanujan when you really need him?!

coming in second to a team of four people including Terrence Tao is really impressive

These videos with Dr. Maynard are great!

Congrats on the Fields Medal, James !!

Congrats James!!!

That factorial proof is so simple yet cool

That is a really cool expression. Mind blowing that one can discover and tinker with something like that. Nicely done!

I went on internet to rest from math, but looks like I won't :D

They split the 10.000. But they split 5000 to each response or 2000 to each person? I think James should get 5000 since he did his own work by himself

I really love your channel. I'm a Biochemist, and most of my life math was just a useful annoyance I had to study for 2 years. I've enjoyed watching your channel so much more than I though I would, and it;s given me a whole new perspective on the meaning of mathematics. Thank-you for doing this.

Big congrats on your fields medal! Well done :-)

Judging by that Rolex, Dr. Maynard, I think I know where the 2000 bucks went

Me pranking a high schooler: “Find a prime number larger than infinity factorial.” 😂

Uploaded 8 minutes ago; video is 9:26 in length; 116 likes... You people have good faith!

I find it absolutely fascinating how Maynard and the other group had completely different approaches to the problem, but got _the exact same_ formula for large prime gaps. Is there some strange connection here? Or was that formula already hypothesised to be the solution, and they simply used different approaches to proving it?

last time I saw so many logs in one place, they were building a cabin!

Would realy like to see more of Dr Maynard!!!!!!!!!!!!!!

Being a mathematician might just be the best job in the world, seeing as how you get addicted to your job... No wonder all of these guys smile all the time :D

No trees were cut down in the making of this video

Love the way he explain it

Why is it that prime numbers, constants and their relations and patterns are so intriguing? I haven't even studied math, had OKish grades in school, but now that i am free of the constraints of school or using math at work it all starts to have such a fascinating glimmer to it. It all started with SDRs and i was fascinated how, with help of i.e. the fourier transformation, you'd be able to extract signal from noise that no human ear could even guess they were there. And you know if you say Fourier, you say "e", "pi", "i"... That was where my jouney began.

You can use Bertrand to show that if x

8:05 "I'm a lumberjack and I'm OK. I sleep all night and work all day!"

0:20 - "... a very high-school argument." I went to Cambridge to do maths, and I didn't see this until my first week there.

It's the first time I've seen James Maynard on a Numberphile video. I look forward to more. Speaking as one of your innumerate viewers, I'd say good job, nice delivery and he look as if he doesn't get out much, like a proper mathematician. Appearances can be deceiving of course.

Very nice!!

Interessting how +6 was used there. I once played around with C a bit and found that the gaps between 2 primes "tend" to be multiples of 6. With "tends" I mean: If you plot the number of primes with gap x vs x you get something similar to a saw. dropping, dropping, oh: X is a multiple of 6: increase a bit again, dropping, dropping, oh: multiple 6: increase etc.

This guy reminds me of fuzzy peaches, sweedish berries, and wine gums for some reason.

How does the Riemann hypothesis play into the problem of large gaps between primes and do the (nontrivial) zeros of the zeta function tell us something about the gaps between primes?

idea for a piece of mathy art: you have your x axis be the increment you increase by and your y axis is the prime you start with, and at every point you color it based on a scale from 0 to the highest number of primes in a row included on the graph. i'd be very interested to see how it turned out, but i don't have the brainpower, patience, or resources to do it myself.

So, the purpose of using logarithms are to deconstruct a variable exponent, right? What was the original equation that required all those logs? That's the link or video I want, Brady!

Please do a video about Maryam Mirzakhani and her work on geometry

I've always been fascinated by the twin prime gaps of the same size such as 199 to 211 and then 211 to 223. Prime number 211 has a gap of 12 in each direction. I wonder if this can be done for every even number.

awesome videos.

You have large gaps between primes is because waves expand as they move from the center. And interference patterns have wide gaps.

Terry Tao & collabs straight up ninja'd James Maynard

My tiny improvement: instead of n! you just need the product of primes

very inspiring..

congratulations for winning the fields medal

What you think of the graph of `f(x) = prime(x)/log(prime(x))/log(x)` ?

If the numbers either side of a primorial Pn# are themselves composite, then the sequence ( Pn# - Pn'+1, ... , Pn# + Pn'-1 ) forms a gap of length 2(Pn' + 1) - 1 [ where Pn, Pn' denotes the nth prime and its successor prime, of course ]. The sequence ( N! + 2, ... , N! + N ) 'only' guarantees a length of N - 1.

Using the simpler expression and simple minded solution, if you wanted to look for an arbitrarily large prime, you could start with X! and then work downward (X!-1) to avoid the known gap.

another maynard!!!

8:16 Hmm, I think James Maynard forgot to square the denominator in this video so the formula differs from both his and the other team's papers.

i was just thinking about length between primes! more specifically if the nth prime over the nth composite converged to a certain point

Please make a video showing how this proof on prime gaps is related to Yitang Zhang's work on prime gaps.

I did just realize that there’s a space of x!-x!/x between every off limit section, which means the gap in which it’s unpredictable grows by a factorial too

So, whose paper got the better bound on consecutive primes? Or did they get the same bound in two different ways?

Looking at numerical experiments, how accurate is the exact bound with respect to what we observe for finite but large x?

Hey, don't be so rough on your formula! logs of logs can take time to calculate, but they make the large numbers significantly smaller! I think it's a great and efficient formula :)

I like how his 0s look like hearts

god I love this channel

8:05 loggers logchamp

A gap of n-1 primes appears much earlier than n!+2. There will also be a gap starting at the least common multiple of 2 through n, plus two, a much smaller number than n!+2.

Who worked longer on his paper? Can you quantify that at all, do you log your time on a certain topic?

Yes !

1:48 I see that long-scale billion there.

SAW MATT PARKER AFTER SCHOOL TODAY AT THE LATYMER SCHOOL, EDMONTON!

2022 Fields medalist!!

Is there any proof for the longest same number gap between consecutive primes? Eg if there was 6 spaces between two primes, then 6 spaces to the next one, then again and again?

I dunno how to express this but... If you take a set of 10, and reduce it to the digits that could possibly be prime (numbers ending with digits 1, 3, 7, and 9). This set of 4 numbers will always be divisible by primes or exponents of primes if they're not prime. I'm not sure if this is known but I assume that someone else must have stumbled across this... but once you know this it seems pretty obvious why there are gaps and how big they should be... they're predictable because they grow as the pool of primes and prime exponents grow and the size of gaps should be fairly easy to figure out from that... likewise where the primes should turn up because it's simply where the multiplies primes and prime exponents don't line up >.> Though I haven't really spent much time on primes and I'm not really good at math, so I think this should be well known but it seems that people don't know it or it just isn't significant...

Just a point of reference. It's pronounced 'Air Dish' I only know this from reading the book 'the man who only loved numbers' a fantastic read by the way.

Was Erdős's prize money a good predictor of how long it would take to solve the problem and therefore how difficult it actually was (although I realize the larger incentive for supposedly harder problems works against this metric)?

So if you put in infinity factorial does that mean there is an infinitely large gap at some point? Implying there are a finite amount of primes on the real number line (the last prime before the infinite gap)?

Whose work was more accurate? Or did both groups end up with the same formula?

other than encryption, is there a use for finding these large primes?

More Cliff Stoll, please.

He had the board but chose the paper.

Is that 100!+x(in which x=1...100) also the first time since 1...100 where you get 100 consecutive divisors for 100 consecutive numbers?

Its amazing how numberphile surpasses vsauce in the number of ways to blow my mind

Can you solve problem 3 on imo 2017? Just curious

MIND THE GAP between the platform and the train!

So we're using "billion" from the long system then? ;)

There should be video on Numberphile about harmonic analysis, Fourier series or wavelets. I mean, isn't it insteresting that we can build almost every function from many tiny parts? Thumbs up if you are also interested!

I am a simple man. I see Paul Erdős - I hit "Like".

This makes me wonder what the ratio of primes/whole numbers is and how that changes over regular intervals in magnitude on the number line.

Thumps up if you find a partialy erased blackboard irrationally annoying.

What's the website he said they publish papers on?

For p>9, don't you just have to check numbers ending in 1, 3, 7, and 9, instead of the other 6 numbers in that group of 10 (e.g. 20-29)? That is because numbers ending (0, 2, 4, 6, 8) are not prime because they are divisible by 2, and those ending in 0 or 5 are divisible by 5. While this is still a linear search, it is better than searching all numbers.

In the prime-gap function at the end of the video, what is the base of the log's?

1:00 1:43: I think everyone's had this thought, then forgot. 2:19 2:39: no even numbers, yes 3:39 4:21 4:49

Where do you get all those logs in number theory?

Consecutive days where solutions are found to a problem relating to consecutive prime numbers are often far apart.

every time i come to this channel i get reminded how stupid i am and i just get depressed.

At the beginning of the video that very wonderful proof not only shows that you can find any arbitrarily large gap in the primes, but also can be used as a formula to give the exact location of those gaps, if you were to work out the extremely large value of the factorials. I don't know much about the proof for the arbitrarily long arithmetic sequences of primes, but does that proof use a similar formula that can also pinpoint the exact location of that arbitrarily long sequence? In other words, could that proof be used to easily find extraordinarily large prime numbers?